Table Of ContentLectu re Notes
in Econom,ics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
Mathematical Economics
223
Walter Trockel
Market Demand
An Analysis of Large Economies
with Non-Convex Preferences
Springer-Verlag
Berlin Heidelberg New York Tokyo 1984
Editorial Board
H. Albach M. Beckmann (Managing Editor) P. Ohrymes
G. Fandel J. Green W Hildenbrand W Krelle (Managing Editor) H.P. KUnzi
G.L. Nemhauser K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W Krelle
Institut fUr Gesellschafts-und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Author
Dr. Walter Trockel
Institut fUr Gesellschafts- und Wirtschaftswissenschaften
Abteilung Wirtschaftstheorie II, Universitat Bonn
Adenauerallee 24-26, 0-5300 Bonn, FRG
ISBN-13: 978-3-540-12881-6 e-ISBN-13: 978-3-642-46488-1
001: 10,1007/978-3-642-46488-1
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Softcover reprint of the hardcover 1st edition 1984
2142/3140-543210
Meinen E.i.telLn
In the non-convex taste situation
there is complete decentralization
since guidance by prices
for agents suffices
due to Smooth~ng by Agg~egat~on
Preface
The present monograph is a synthesis of what has been contributed
during the last decade to the analysis of market demand in large econ
omies where consumers may have non-convex preference relations. Al
though research in this field has not yet come to an end there exists
a variety of interesting results, established in different frameworks
by means of different conceptual and formal tools.
It is my aim to givea comprehensive treatment of the existing lit
erature including my own contributions.
In working out differences and interrelations of the various ap
proaches I adopted and modified several of the original results.
My desire to present the problem and the methods by which it has
been treated in such a way, that also non-specialists can follow, con
flicted sometimes with the inevitable complexity of tools to be used.
Therefore, I decided to give enough room to the introductory and prepa
ratory part of this work. This part consists of the introduction and
of the first four chapters.
The main part of the present analysis consists of chapters 5 to 7.
The heart of the whole work is chapter 6 which presents the "ergodic
approach" to smoothing demand by aggregation, jointly developed by
Egbert and Hildegard Dierker and myself. In chapter 8 mathematical
notions and facts are collected in such a way as to make the book to
a certain extent selfcontained. An exception is the treatment of catas
trophes which is very sketchy and is intended only as a first source
of quick information.
Let me conclude with some acknowledgements. During the last few
years I had the opportunity to speak with all of the people who have
contributed to the field dealt with in the present book, since they
were in Bonn either permanently or as visitors. In particular, I want
to thank Gerard Debreu and Werner Hildenbrand, the initiators of re
search in this field, for permanent support and encouragement during
my work and to Andreu Mas-Colell for inviting me to Berkeley and for
the valuable discussions with him. My thank is also due to Dieter Sonder
mann and Hans Wiesmeth for conversations which helped to deepen my
insight into their contributions to the parametric approach.
My special gratitude belongs to Egbert Dierker and Hildegard Dierker.
Not only are main parts of the present work based on my joint work with
them. Also, what regards my single contributions and the preparation
of the present work, they were always available as critical advisers
who helped by constructive comments.
While I have been working on aggregate demand I profitted from con
versations with several friends and colleagues. In particular I am in
depted to Beth Allen and Wilhelm Neuefeind who also read and commented
parts of the book, and to Truman Bewley, Hans Haller, Simcha Knif, David
Rand, Jacob Weinberg, and Akira Yamazaki.
To Peter Dax, Gert Schrader and Rolf-Peter Schneider I am grateful
for their skillful typing.
Finally, I gratefully acknowledge the generous grant of the Deutsche
Forschungsgemeinschaft which made it possible for me to visit the Uni
versity of California and to write the present monograph.
Contents
Introduction
Demand and Equilibria 6
1.1 Introduction 6
1.2 Individual demand 7
1.3 Market demand 11
1.4 Equilibria 17
1.5 Concluding remarks 21
2 Smoothing Demand by Aggregation 23
2.1 Many different consumers 23
2.2 Convexifying by averaging 30
2.3 Concluding remarks on diversification 32
3 Spaces of Preferences and Large Economies 34
3.1 Introduction 34
3.2 Preference relations 35
3.3 Spaces of preference relations 42
3.4 Large economies 48
4 Heuristic Remarks on Some Basic Concepts 53
4.1 Genericity 53
4.2 Stability 56
4.3 Equal distribution 59
4.4 The law of large numbers 61
5 The Parametric Approach 65
5.1 Introduction 65
5.2 Dispersion of tastes 67
5.3 Continuous mean demand 71
5.4 Differentiable mean demand 88
5.5 Concluding remarks 94
6 The Ergodic Approach 98
6.1 Introduction 98
6.2 G-dispersion 104
6.3 Continuous mean demand 11 8
6.4 Differentiable mean demand 124
6.5 Concluding remarks 157
VIII
7 Other Approaches 161
7.1 Dispersion of unit normals 1 61
7.2 Infinite dimensional linear spaces of prefer
ences 166
7.3 Random individual demand 171
7.4 The generic approach 175
8 Mathematical Tools 179
8.1 Miscellany 179
8.2 Differential topology 184
8.3 Measure theory 189
Notation 196
References 199
Introduction
In the present work I shall analyze a problem of economic theory.
Its most general version is relevant in several areas of social
sciences and can be described as follows:
Consider a group of individuals and for each individual a deci
sion problem, which depends on a common parameter. If the results of
the individual decision processes are aggregated, what can be said
about the aggregate outcome and its dependence on the parameter?
In general the individual decision problems cannot be expected to
have unique solutions. Hence it is of particular interest to know con
ditions under which there is for every parameter a unique aggregate
outcome. Moreover, in the case of uniqueness, it is important to know
whether the aggregate outcome, as a function of the common parameter,
has a special structure.
Typical decision problems in economics are optimization problems.
In particular, the "Utility Maximization Hypothesis" is basic for the
theory of consumer's behavior.
In this context the parameters are the prices of the various com
modities. The set of solutions for an individual decision problem at
a certain parameter (i.e., price system) is then the individual's de
mand set at a price system. In other words, it is the set of optimal
commodity bundles in the budget set, where the budget set is deter
mined by the prevailing price system and the individual's initial
holding. The non-uniqueness of individual demand sets is caused in
this context by the lack of strict convexity of preference relations.
2
This is illustrated for a non-convex preference in Figure 1.1 and for
a convex, but not strictly convex preference in Figure 1.2.
Figure 1.1 Figure 1.2
In the first case there are two optimal elements, x and y, in the
second case there is a whole segment, [x, yl, of optimal elements.
The aggregate outcome in this framework is the sum of individual
demand sets, the market demand.
In the following I shall confine myself to an analysis of this
specific version of the general problem stated at the beginning. Ac
cordingly, in the framework of Walrasian equilibrium theory I shall
be concerned with the question, when market demand is uniquely deter
mined at every price system, and, if so, when it is a well behaved
(continuous; continuously differentiable) function of prices.
Clearly, a summation of finitely many sets can impossibly result
in a single point unless each of these sets is a singleton. However,
if the number of individuals is very large, and if summation is re
placed by averaging, i.e., if one replaces total demand by total de
~per capita, then the situation is different. Indeed, if, at a cer
tainprice system, most of the consumers happen to have single-valued
demands, then the mean demand, i.e., the total demand per capita be
comes a small set. The appropriate way to m?del such a situation is by
means of a continuum of consumers. Averaging is formalized then as in
tegration with respect to a probability measure.
It is obvious that the mere presence of many consumers in the mar
ket cannot suffice to yield uniqueness of market demand. Consid~r many
